{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:25Z","timestamp":1753893805562,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over ${\\Bbb Z}$ and $x_m, m \\in {\\Bbb Z}$ be rational functions defined by $x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k \\in {\\Bbb Z}$, $x_k$ can be expressed as a substraction-free Laurent polynomial in ${\\Bbb Z}[x_m^{\\pm 1},x_{m+1}^{\\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.<\/jats:p>","DOI":"10.37236\/187","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:27:39Z","timestamp":1578716859000},"source":"Crossref","is-referenced-by-count":2,"title":["Positivity in Coefficient-Free Rank Two Cluster Algebras"],"prefix":"10.37236","volume":"16","author":[{"given":"G.","family":"Dupont","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,8,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r98\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r98\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:41:58Z","timestamp":1579315318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r98"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/187","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,8,7]]},"article-number":"R98"}}